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Journal articles on the topic 'Time-Scales calculus'

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Author: Grafiati

Published: 25 May 2024

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1

Khan, A. R., F. Mehmood, and M. A. Shaikh. "Обобщение неравенств Островского на временных шкалах." Владикавказский математический журнал 25, no. 3 (September 25, 2023): 98–110. http://dx.doi.org/10.46698/q4172-3323-1923-j.

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The idea of time scales calculus’ theory was initiated and introduced by Hilger (1988) in his PhD thesis order to unify discret and continuous analysis and to expend the discrete and continous theories to cases ``in between''. Since then, mathematical research in this field has exceeded more than 1000 publications and a lot of applications in the fields of science, i.e., operations research, economics, physics, engineering, statistics, finance and biology. Ostrowski proved an inequality to estimate the absolute deviation of a differentiable function from its integral mean. This result was obtained by applying the Montgomery identity. In the present paper we derive a generalization of the Montgomery identity to the various time scale versions such as discrete case, continuous case and the case of quantum calculus, by obtaining this generalization of Montgomery identity we would prove our results about the generalization of the Ostrowski inequalities (without weighted case) to the several time scales such as discrete case, continuous case and the case of quantum calculus and recapture the several published results of different authors of various papers and thus unify corresponding discrete version and continuous version. Similarly we would also derive our results about the generalization of the Ostrowski inequalities (weighted case) to the different time scales such as discrete case and continuous case and recapture the different published results of several authors of various papers and thus unify corresponding discrete version and continuous version. Moreover, we would use our obtained results (without weighted case) to the case of quantum calculus.

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2

Torrest, Delfim F. M. "The variational calculus on time scales." International Journal for Simulation and Multidisciplinary Design Optimization 4, no. 1 (January 2010): 11–25. http://dx.doi.org/10.1051/ijsmdo/2010003.

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3

Yaslan, İsmail. "Beta-Fractional Calculus on Time Scales." Journal of Fractional Calculus and Nonlinear Systems 4, no. 2 (December 27, 2023): 48–60. http://dx.doi.org/10.48185/jfcns.v4i2.877.

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4

Sahir, Muhammad Jibril Shahab. "Uniformity of dynamic inequalities constituted on time Scales." Engineering and Applied Science Letters 3, no. 4 (October 24, 2020): 19–27. http://dx.doi.org/10.30538/psrp-easl2020.0048.

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In this article, we present extensions of some well-known inequalities such as Young's inequality and Qi's inequality on fractional calculus of time scales. To find generalizations of such types of dynamic inequalities, we apply the time scale Riemann-Liouville type fractional integrals. We investigate dynamic inequalities on delta calculus and their symmetric nabla results. The theory of time scales is utilized to combine versions in one comprehensive form. The calculus of time scales unifies and extends some continuous forms and their discrete and quantum inequalities. By applying the calculus of time scales, results can be generated in more general form. This hybrid theory is also extensively practiced on dynamic inequalities.

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5

Malinowska, Agnieszka B., and Natália Martins. "The Second Noether Theorem on Time Scales." Abstract and Applied Analysis 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/675127.

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We extend the second Noether theorem to variational problems on time scales. As corollaries we obtain the classical second Noether theorem, the second Noether theorem for theh-calculus and the second Noether theorem for theq-calculus.

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6

Sahir, Muhammad Jibril Shahab. "Coordination of Classical and Dynamic Inequalities Complying on Time Scales." European Journal of Mathematical Analysis 3 (February 3, 2023): 12. http://dx.doi.org/10.28924/ada/ma.3.12.

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In this research article, we present extensions of some classical inequalities such as Schweitzer, Pólya–Szegö, Kantorovich and Greub–Rheinboldt inequalities of fractional calculus on time scales. To investigate generalizations of such types of classical inequalities, we use the time scales Riemann–Liouville type fractional integrals. We explore dynamic inequalities on delta calculus and their symmetric nabla versions. A time scale is an arbitrary nonempty closed subset of the real numbers. The theory of time scales is applied to combine results in one comprehensive form. The calculus of time scales unifies and extends continuous versions and their discrete and quantum analogues. By using the calculus of time scales, results are presented in more general form. This hybrid theory is also widely applied on dynamic inequalities.

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7

Ganie, Javid Ahmad, and Renu Jain. "THE SUMUDU TRANSFORM ON DISCRETE TIME SCALES." Jnanabha 51, no. 02 (2021): 58–67. http://dx.doi.org/10.58250/jnanabha.2021.51208.

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The study of dynamic equations on time scale is an area of mathematics that has recently received a lot of attention. This type of calculus has been created in order to unify the study of discrete and continuous analysis. Integral transforms play a crucial role in analysis in solving differential and difference equations. In this paper, we introduce the Sumudu transform on two different time scales by using the theory of time scale calculus. Finally, we employ these definitions to derive the results like, convolution, delay(shift) and inversion on these discrete time scales. Further, these results coincide with those in continuous case.

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8

Sahir, M. J. S. "Объединение классических и динамических неравенств, возникающих при анализе временных масштабов." Вестник КРАУНЦ. Физико-математические науки, no. 4 (December 29, 2020): 26–36. http://dx.doi.org/10.26117/2079-6641-2020-33-4-26-36.

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In this paper, we present an extension of dynamic Renyi’s inequality on time scales by using the time scale Riemann–Liouville type fractional integral. Furthermore, we find generalizations of the well–known Lyapunov’s inequality and Radon’s inequality on time scales by using the time scale Riemann–Liouville type fractional integrals. Our investigations unify and extend some continuous inequalities and their corresponding discrete analogues. В этой статье мы представляем расширение динамического неравенства Реньи на шкалы времени с помощью дробного интеграла типа Римана-Лиувилля. Кроме того, мы находим обобщения хорошо известного неравенства Ляпунова и неравенства Радона на шкалах времени с помощью дробных интегралов типа Римана-Лиувилля на шкале. Наши исследования объединяют и расширяют некоторые непрерывные неравенства и соответствующие им дискретные аналоги.

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9

Zhao, Dafang, and Tongxing Li. "On conformable delta fractional calculus on time scales." Journal of Mathematics and Computer Science 16, no. 03 (September 15, 2016): 324–35. http://dx.doi.org/10.22436/jmcs.016.03.03.

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10

Seiffertt, John. "Adaptive Resonance Theory in the time scales calculus." Neural Networks 120 (December 2019): 32–39. http://dx.doi.org/10.1016/j.neunet.2019.08.010.

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11

Benkhettou, Nadia, Salima Hassani, and Delfim F. M. Torres. "A conformable fractional calculus on arbitrary time scales." Journal of King Saud University - Science 28, no. 1 (January 2016): 93–98. http://dx.doi.org/10.1016/j.jksus.2015.05.003.

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12

Fard, Omid Solaymani, and Tayebeh A. Bidgoli. "Calculus of fuzzy functions on time scales (I)." Soft Computing 19, no. 2 (March 12, 2014): 293–305. http://dx.doi.org/10.1007/s00500-014-1252-6.

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13

Bohner, Martin, and Gusein Sh Guseinov. "Double integral calculus of variations on time scales." Computers & Mathematics with Applications 54, no. 1 (July 2007): 45–57. http://dx.doi.org/10.1016/j.camwa.2006.10.032.

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14

Ufuktepe, Ünal, and Sinan Kapçak. "Unification of calculus on Time Scales with mathematica." Applied Mathematics and Computation 218, no. 3 (October 2011): 1102–6. http://dx.doi.org/10.1016/j.amc.2011.03.030.

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15

Rezk, Haytham M., Mahmoud I. Mohammed, Oluwafemi Samson Balogun, and Ahmed I. Saied. "Exploring Generalized Hardy-Type Inequalities via Nabla Calculus on Time Scales." Symmetry 15, no. 9 (August 27, 2023): 1656. http://dx.doi.org/10.3390/sym15091656.

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In this research, we aim to explore generalizations of Hardy-type inequalities using nabla Hölder’s inequality, nabla Jensen’s inequality, chain rule on nabla calculus and leveraging the properties of convex and submultiplicative functions. Nabla calculus on time scales provides a unified framework that unifies continuous and discrete calculus, making it a powerful tool for studying various mathematical problems on time scales. By utilizing this approach, we seek to extend Hardy-type inequalities beyond their classical continuous or discrete settings to a more general time scale domain. As specific instances of our discoveries, we have the integral inequalities previously established in the existing literature.

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16

Zakarya, Mohammed, A. I. Saied, Maha Ali, Haytham M. Rezk, and Mohammed R. Kenawy. "Novel Integral Inequalities on Nabla Time Scales with C-Monotonic Functions." Symmetry 15, no. 6 (June 12, 2023): 1248. http://dx.doi.org/10.3390/sym15061248.

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Through the paper, we present several inequalities involving C-monotonic functions with C≥1, on nabla calculus via time scales. It is known that dynamic inequalities generate many different inequalities in different calculus. The main results will be proved by applying the chain rule formula on nabla calculus. As a special case for our results, when T=R, we obtain the continuous analouges of inequalities that had previously been proved in the literature. When T=N, the results, to the best of the authors’ knowledge, are essentially new. Symmetrical properties of C-monotonic functions are critical in determining the best way to solve inequalities.

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17

Nosheen, Ammara, Aneeqa Aslam, Khuram Ali Khan, Khalid Mahmood Awan, and Hamid Reza Moradi. "Multivariate Dynamic Sneak-Out Inequalities on Time Scales." Journal of Mathematics 2021 (May 26, 2021): 1–17. http://dx.doi.org/10.1155/2021/9978050.

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In this study, we extend some “sneak-out” inequalities on time scales for a function depending on more than one parameter. The results are proved by using the induction principle and time scale version of Minkowski inequalities. In seeking applications, these inequalities are discussed in classical, discrete, and quantum calculus.

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18

Martins, Natália, and Delfim F. M. Torres. "Calculus of variations on time scales with nabla derivatives." Nonlinear Analysis: Theory, Methods & Applications 71, no. 12 (December 2009): e763-e773. http://dx.doi.org/10.1016/j.na.2008.11.035.

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19

Seiffertt, J., and D. C. Wunsch. "Backpropagation and Ordered Derivatives in the Time Scales Calculus." IEEE Transactions on Neural Networks 21, no. 8 (August 2010): 1262–69. http://dx.doi.org/10.1109/tnn.2010.2050332.

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20

Smoljak Kalamir, Ksenija. "New Diamond-α Steffensen-Type Inequalities for Convex Functions over General Time Scale Measure Spaces." Axioms 11, no. 7 (July 1, 2022): 323. http://dx.doi.org/10.3390/axioms11070323.

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In this paper, we extend some Steffensen-type inequalities to time scales by using the diamond-α-dynamic integral. Further, we prove some new Steffensen-type inequalities for convex functions utilizing positive σ-finite measures in time scale calculus. Moreover, as a special case, we obtain these inequalities for the delta and the nabla integral. By using the relation between calculus on time scales T and differential calculus on R, we obtain already-known Steffensen-type inequalities.

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21

Gogoi, Bikash, Utpal Kumar Saha, Bipan Hazarika, Delfim F. M. Torres, and Hijaz Ahmad. "Nabla Fractional Derivative and Fractional Integral on Time Scales." Axioms 10, no. 4 (November 24, 2021): 317. http://dx.doi.org/10.3390/axioms10040317.

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In this paper, we introduce the nabla fractional derivative and fractional integral on time scales in the Riemann–Liouville sense. We also introduce the nabla fractional derivative in Grünwald–Letnikov sense. Some of the basic properties and theorems related to nabla fractional calculus are discussed.

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22

Kaymakçalan, Billur. "Lyapunov stability theory for dynamic systems on time scales." Journal of Applied Mathematics and Stochastic Analysis 5, no. 3 (January 1, 1992): 275–81. http://dx.doi.org/10.1155/s1048953392000224.

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By use of the necessary calculus and the fundamental existence theory for dynamic systems on time scales, in this paper, we develop Lyapunov's second method in the framework of general comparison principle so that one can cover and include several stability results for both types of equations at the same time.

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23

El-Deeb, Ahmed A., Samer D. Makharesh, and Barakah Almarri. "Some New Inverse Hilbert Inequalities on Time Scales." Symmetry 14, no. 11 (October 25, 2022): 2234. http://dx.doi.org/10.3390/sym14112234.

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Several inverse integral inequalities were proved in 2004 by Yong. It is our aim in this paper to extend these inequalities to time scales. Furthermore, we also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases. Our results are proved using some algebraic inequalities, inverse Hölder’s inequality and inverse Jensen’s inequality on time scales. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

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24

Zakarya, Mohammed, Ghada AlNemer, Ahmed I. Saied, Roqia Butush, Omar Bazighifan, and Haytham M. Rezk. "Generalized Inequalities of Hilbert-Type on Time Scales Nabla Calculus." Symmetry 14, no. 8 (July 24, 2022): 1512. http://dx.doi.org/10.3390/sym14081512.

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In this paper, we prove some new generalized inequalities of Hilbert-type on time scales nabla calculus by applying Hölder’s inequality, Young’s inequality, and Jensen’s inequality. Symmetrical properties play an essential role in determining the correct methods to solve inequalities.

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25

Aly, Elkhateeb S., Y. A. Madani, F. Gassem, A. I. Saied, H. M. Rezk, and Wael W. Mohammed. "Some dynamic Hardy-type inequalities with negative parameters on time scales nabla calculus." AIMS Mathematics 9, no. 2 (2024): 5147–70. http://dx.doi.org/10.3934/math.2024250.

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<abstract><p>In this paper, we establish some new dynamic Hardy-type inequalities with negative parameters on time scales nabla calculus by applying the reverse H ölder's inequality, integration by parts, and chain rule on time scales nabla calculus. As special cases of our results (when $ \mathbb{ T = R} $), we get the continuous analouges of inequalities proven by Benaissa and Sarikaya, and when $ \mathbb{T = N}_{0} $, the results to the best of the authors' knowledge are essentially new.</p></abstract>

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26

PELEN, Neslihan Nesliye, and Zeynep KAYAR. "Falling Body Motion in Time Scale Calculus." Gazi University Journal of Science Part A: Engineering and Innovation 11, no. 1 (March 21, 2024): 210–24. http://dx.doi.org/10.54287/gujsa.1427944.

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The falling body problem for different time scales, such as ℝ, ℤ, hℤ, qℕ0, ℙc,d is the subject of this study. To deal with this problem, we use time-scale calculus. Time scale dynamic equations are used to define the falling body problem. The exponential time scale function is used for the solutions of these problems. The solutions of the falling body problem in each of these time scales are found. Moreover, we also test our mathematical results with numerical simulations.

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27

Zakarya, Mohammed, Mohamed Altanji, Ghada AlNemer, Hoda A. Abd El-Hamid, Clemente Cesarano, and Haytham M. Rezk. "Fractional Reverse Coposn’s Inequalities via Conformable Calculus on Time Scales." Symmetry 13, no. 4 (March 25, 2021): 542. http://dx.doi.org/10.3390/sym13040542.

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This paper provides novel generalizations by considering the generalized conformable fractional integrals for reverse Copson’s type inequalities on time scales. The main results will be proved using a general algebraic inequality, chain rule, Hölder’s inequality, and integration by parts on fractional time scales. Our investigations unify and extend some continuous inequalities and their corresponding discrete analogues. In addition, when α = 1, we obtain some well-known time scale inequalities due to Hardy, Copson, Bennett, and Leindler inequalities.

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28

Akgandüller, Ömer, and Sibel Paşalı Atmaca. "Discrete Normal Vector Field Approximation via Time Scale Calculus." Applied Mathematics and Nonlinear Sciences 5, no. 1 (March 31, 2020): 349–60. http://dx.doi.org/10.2478/amns.2020.1.00033.

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AbstractThe theory of time scales calculus have long been a subject to many researchers from different disciplines. Beside the unification and the extension aspects of the theory, it emerge as a powerful tool for mimetic discretization process. In this study, we present a framework to find normal vector fields of discrete point sets in ℝ3 by using symmetric differential on time scales. A surface parameterized by the tensor product of two time scales can be analogously expressed as the vertex set of non-regular rectangular grids. If the time scales are dense, then the discrete grid structure vanishes. If the time scales are isolated, then the further geometric analysis can be executed by using symmetric dynamic differential. Moreover, we present an algorithmic procedure to determine the symmetric dynamic differential structure on the neighborhood of points in surfaces. Our results indicate that the method we present has good approximation to unit normal vector fields of parameterized surfaces rather than the Delaunay triangulation for some points.

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29

Shen, Jian-Mei, Saima Rashid, Muhammad Aslam Noor, Rehana Ashraf, and Yu-Ming Chu. "Certain novel estimates within fractional calculus theory on time scales." AIMS Mathematics 5, no. 6 (2020): 6073–86. http://dx.doi.org/10.3934/math.2020390.

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30

Özkan, Umut Mutlu, and Billûr Kaymakçalan. "Basics of diamond-α partial dynamic calculus on time scales." Mathematical and Computer Modelling 50, no. 9-10 (November 2009): 1253–61. http://dx.doi.org/10.1016/j.mcm.2009.01.007.

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31

Anastassiou, George A. "Principles of delta fractional calculus on time scales and inequalities." Mathematical and Computer Modelling 52, no. 3-4 (August 2010): 556–66. http://dx.doi.org/10.1016/j.mcm.2010.03.055.

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32

Agarwal, Ravi P., and Martin Bohner. "Basic Calculus on Time Scales and some of its Applications." Results in Mathematics 35, no. 1-2 (March 1999): 3–22. http://dx.doi.org/10.1007/bf03322019.

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33

Anastassiou, George A. "Foundations of nabla fractional calculus on time scales and inequalities." Computers & Mathematics with Applications 59, no. 12 (June 2010): 3750–62. http://dx.doi.org/10.1016/j.camwa.2010.03.072.

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34

Bourdin, Loïc. "Nonshifted calculus of variations on time scales with ∇-differentiable σ." Journal of Mathematical Analysis and Applications 411, no. 2 (March 2014): 543–54. http://dx.doi.org/10.1016/j.jmaa.2013.10.013.

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35

Cheng, Quanxin, and Jinde Cao. "Global Synchronization of Complex Networks with Discrete Time Delays on Time Scales." Discrete Dynamics in Nature and Society 2011 (2011): 1–19. http://dx.doi.org/10.1155/2011/287670.

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This paper studies the global synchronization problem for a class of complex networks with discrete time delays. By using the theory of calculus on time scales, the properties of Kronecker product, and Lyapunov method, some sufficient conditions are obtained to ensure the global synchronization of the complex networks with delays on time scales. These sufficient conditions are formulated in terms of linear matrix inequalities (LMIs). The main contribution of the result is that the global synchronization problems with both discrete time and continuous time are unified under the same framework.

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36

Li, Yongkun, and Chao Wang. "Almost Periodic Functions on Time Scales and Applications." Discrete Dynamics in Nature and Society 2011 (2011): 1–20. http://dx.doi.org/10.1155/2011/727068.

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We first propose the concept of almost periodic time scales and then give the definition of almost periodic functions on almost periodic time scales, then by using the theory of calculus on time scales and some mathematical methods, some basic results about almost periodic differential equations on almost periodic time scales are established. Based on these results, a class of high-order Hopfield neural networks with variable delays are studied on almost periodic time scales, and some sufficient conditions are established for the existence and global asymptotic stability of the almost periodic solution. Finally, two examples and numerical simulations are presented to illustrate the feasibility and effectiveness of the results.

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37

Rezk, Haytham M., Ahmed I. Saied, Maha Ali, Belal A. Glalah, and Mohammed Zakarya. "Novel Hardy-Type Inequalities with Submultiplicative Functions on Time Scales Using Delta Calculus." Axioms 12, no. 8 (August 16, 2023): 791. http://dx.doi.org/10.3390/axioms12080791.

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In this study, we apply Hölder’s inequality, Jensen’s inequality, chain rule and the properties of convex functions and submultiplicative functions to develop an innovative category of dynamic Hardy-type inequalities on time scales delta calculus. A time scale, denoted by T, is any closed nonempty subset of R. In time scale calculus, results are unified and extended. As particular cases of our findings (when T=R), we have the continuous analogues of inequalities established in some the literature. Furthermore, we can find other inequalities in different time scales, such as T=N, which, to the best of the authors’ knowledge, is a largely novel conclusion.

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38

Dryl, Monika, and Delfim F. M. Torres. "Necessary Condition for an Euler-Lagrange Equation on Time Scales." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/631281.

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We prove a necessary condition for a dynamic integrodifferential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. An example of a second order dynamic equation, which is not an Euler-Lagrange equation on an arbitrary time scale, is given.

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39

MEHMOOD, FARAZ, and AKHMADJON SOLEEV. "NEW GENERALIZATION OF OSTROWSKI-GRÜSS LIKE INEQUALITY ON TIME SCALES." 2022-yil 3-son (133/1) ANIQ FANLAR SERIYASI 1, no. 1 (February 14, 2023): 1–8. http://dx.doi.org/10.59251/2181-1296.v1.1.1.

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In the present paper we obtain the generalized inequality of Ostrowski-Grüss like on time scales using three steps kernel and recapture several established results of different authors of different papers and thus unify corresponding discrete and continuous versions. Further that we apply our obtained consequence to the case of quantum calculus.

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40

M E H M O O D, F. A. R. A. Z., and A. K. H. M. A. D. J. O. N. S O L E E V. "NEW GENERALIZATION OF OSTROWSKI-GRÜSS LIKE INEQUALITY ON TIME SCALES." 2022-yil, 3-son (133/1) ANIQ FANLAR SERIYASI 1, no. 1 (February 20, 2023): 1–11. http://dx.doi.org/10.59251/2181-1296.v1.1.1894.

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In the present paper we obtain the generalized inequality of Ostrowski-Grüss like on time scales using three steps kernel and recapture several established results of different authors of different papers and thus unify corresponding discrete and continuous versions. Further that we apply our obtained consequence to the case of quantum calculus.

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41

M E H M O O D, F. A. R. A. Z., and A. K. H. M. A. D. J. O. N. S O L E E V. "NEW GENERALIZATION OF OSTROWSKI-GRÜSS LIKE INEQUALITY ON TIME SCALES." 2022-yil, 3-son (133/1) ANIQ FANLAR SERIYASI 1, no. 1 (February 20, 2023): 1–11. http://dx.doi.org/10.59251/2181-1296.2023.v1.1.1894.

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In the present paper we obtain the generalized inequality of Ostrowski-Grüss like on time scales using three steps kernel and recapture several established results of different authors of different papers and thus unify corresponding discrete and continuous versions. Further that we apply our obtained consequence to the case of quantum calculus.

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42

Yan, Wu, and Fu Jing-Li. "Noether’s theorems of variable mass systems on time scales." Applied Mathematics and Nonlinear Sciences 3, no. 1 (May 29, 2018): 229–40. http://dx.doi.org/10.21042/amns.2018.1.00017.

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AbstractThis paper deals with the Noether’s theory for variable mass system on time scales. The calculus on time scales unifies and extends variable mass system continuous model and discrete model into a single theory. Firstly, Hamilton’s principle of the variable mass system on time scales is given. Secondly, based on the quasi-invariance of the Hamilton’s action under a group of infinitesimal transformations, Noether’s theorem and its inverse theorem of the variable mass system on time scales are presented. Finally, two examples are given to illustrate the applications of the results.

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43

Cai, Jinxiang, Zhenkun Huang, and Honghua Bin. "Exponential Stability of Periodic Solution to Wilson-Cowan Networks with Time-Varying Delays on Time Scales." Advances in Artificial Neural Systems 2014 (April 2, 2014): 1–10. http://dx.doi.org/10.1155/2014/750532.

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We present stability analysis of delayed Wilson-Cowan networks on time scales. By applying the theory of calculus on time scales, the contraction mapping principle, and Lyapunov functional, new sufficient conditions are obtained to ensure the existence and exponential stability of periodic solution to the considered system. The obtained results are general and can be applied to discrete-time or continuous-time Wilson-Cowan networks.

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44

El-Deeb, Ahmed A., Dumitru Baleanu, and Jan Awrejcewicz. "(γ,a)-Nabla Reverse Hardy–Hilbert-Type Inequalities on Time Scales." Symmetry 14, no. 8 (August 17, 2022): 1714. http://dx.doi.org/10.3390/sym14081714.

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In this article, using a (γ,a)-nabla conformable integral on time scales, we study several novel Hilbert-type dynamic inequalities via nabla time scales calculus. Our results generalize various inequalities on time scales, unifying and extending several discrete inequalities and their corresponding continuous analogues. We say that symmetry plays an essential role in determining the correct methods with which to solve dynamic inequalities.

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45

Hanif, Usama, Ammara Nosheen, Rabia Bibi, Khuram Ali Khan, and Hamid Reza Moradi. "Some Hardy-Type Inequalities for Superquadratic Functions via Delta Fractional Integrals." Mathematical Problems in Engineering 2021 (May 28, 2021): 1–14. http://dx.doi.org/10.1155/2021/9939468.

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In this paper, Jensen and Hardy inequalities, including Pólya–Knopp type inequalities for superquadratic functions, are extended using Riemann–Liouville delta fractional integrals. Furthermore, some inequalities are proved by using special kernels. Particular cases of obtained inequalities give us the results on time scales calculus, fractional calculus, discrete fractional calculus, and quantum fractional calculus.

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46

Hu, Meng, and Lili Wang. "Unique Existence Theorem of Solution of Almost Periodic Differential Equations on Time Scales." Discrete Dynamics in Nature and Society 2012 (2012): 1–8. http://dx.doi.org/10.1155/2012/240735.

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By using the theory of calculus on time scales andM-matrix theory, the unique existence theorem of solution of almost periodic differential equations on almost periodic time scales is established. The result can be used to a large of dynamic systems.

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47

Rezk, Haytham M., Juan E. Nápoles Valdés, Maha Ali, Ahmed I. Saied, and Mohammed Zakarya. "Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities." Mathematics 12, no. 1 (December 28, 2023): 104. http://dx.doi.org/10.3390/math12010104.

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In this article, we establish some new generalized inequalities of the Hilbert-type on time scales’ delta calculus, which can be considered similar to formulas for inequalities of Hilbert type. The major innovation point is to establish some dynamic inequalities of the Hilbert-type on time scales’ delta calculus for delta differentiable functions of one variable and two variables. In this paper, we use the condition aj(sj)=0 and aj(sj,zj)=aj(wj,nj)=0, ∀j=1,2,…,n. These inequalities will be proved by applying Hölder’s inequality, the chain rule on time scales, and the mean inequality. As special cases of our results (when T=N and T=R), we obtain the discrete and continuous inequalities. Also, we can obtain other inequalities in different time scales, like T=qZ−, q>1.

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48

Nosheen, Ammara, Huma Akbar, Maroof Ahmad Sultan, Jae Dong Chung, and Nehad Ali Shah. "Hardy–Leindler, Yang and Hwang Inequalities for Functions of Several Variables via Time Scale Calculus." Symmetry 14, no. 4 (April 12, 2022): 802. http://dx.doi.org/10.3390/sym14040802.

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In this paper, Hardy–Leindler, Hardy–Yang and Hwang type inequalities are extended on time scales calculus. These extensions are depending upon use of symmetric multiple delta integrals. The target is achieved by utilizing some inequalities in literature along with mathematical induction principle and Fubini’s theorem on time scales. The obtained inequalities are discussed in discrete, continuous and quantum calculus in search of applications. Particular cases of proved results include Hardy, Copson, Hardy–Littlewood, Levinson and Bennett-type inequalities for symmetric sums.

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El-Deeb, A. A., H. A. Elsennary, and Eze R. Nwaeze. "Generalized Weighted Ostrowski, Trapezoid and Grüss Type Inequalities on Time Scales." Fasciculi Mathematici 60, no. 1 (June 1, 2018): 123–44. http://dx.doi.org/10.1515/fascmath-2018-0008.

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Abstract In this article, using two parameters, we obtain generalizations of a weighted Ostrowski type inequality and its companion inequalities on an arbitrary time scale for functions whose first delta derivatives are bounded. Our work unifies the continuous and discrete versions and can also be applied to the quantum calculus case.

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50

Ahmed, Ahmed M., Ahmed I. Saied, Maha Ali, Mohammed Zakarya, and Haytham M. Rezk. "Generalized Dynamic Inequalities of Copson Type on Time Scales." Symmetry 16, no. 3 (March 1, 2024): 288. http://dx.doi.org/10.3390/sym16030288.

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This paper introduces novel generalizations of dynamic inequalities of Copson type within the framework of time scales delta calculus. The proposed generalizations leverage mathematical tools such as Hölder’s inequality, Minkowski’s inequality, the chain rule on time scales, and the properties of power rules on time scales. As special cases of our results, particularly when the time scale T equals the real line (T=R), we derive some classical continuous analogs of previous inequalities. Furthermore, when T corresponds to the set of natural numbers including zero (T=N0), the obtained results, to the best of the authors’ knowledge, represent innovative contributions to the field.

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